A Compact Procedure for Discretization of the Anisotropic Diffusion Operator

This article presents a conservative finite-volume-based scheme for discretization of the anisotropic diffusion operator, which is simple to implement and numerically conservative. The procedure is an extension of a standard method used for discretization of the isotropic diffusion operator in finite-volume methods. Consequently, the technique can be easily implemented in codes originally developed to solve isotropic diffusion problems. The method is applied in an unstructured finite-volume solver and assessed by solving the following four problems: (1) steady conduction in an anisotropic block, (2) anisotropic conduction with a source term, (3) anisotropic conduction in a hollow block, and (4) heterogeneous anisotropic conduction with embedded point heat sources. Calculations are performed for a wide range of parameters using triangular/quadrilateral cells on grid systems with sizes ranging between 102 and 3105 control volumes, and convergence is accelerated through the use of an algebraic multigrid method. Converged solutions are obtained for all problems and on all grid systems used, and the results are in excellent agreement with the exact analytical solutions.